A056241 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-14 - 12\,x + 17\,n + 16\,n x - 3\,n^{2} - 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-30 - 46\,x - 18\,x^{2} + 12\,n + 20\,n x + 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(-12 - 24\,x - 16\,x^{2} - 4\,x^{3}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	5	5	1													11
3	14	21	9	1												45
4	41	82	55	14	1											193
5	122	305	280	115	20	1										843
6	365	1095	1278	731	210	27	1									3707
7	1094	3829	5425	3990	1617	350	35	1								16341
8	3281	13124	21874	19688	10347	3192	546	44	1							72097
9	9842	44289	84888	90426	58518	23571	5796	810	54	1						318195
10	29525	147625	319855	393670	302945	151015	48765	9870	1155	65	1					1404491
11	88574	487157	1177297	1644159	1467400	873257	350471	93621	15972	1595	77	1				6199581
12	265721	1594326	4251537	6643030	6748218	4672278	2246113	748506	169290	24794	2145	90	1			27366049
13	797162	5181553	15112864	26123669	29766958	23519418	13190424	5280717	1494922	291434	37180	2821	104	1		120799227
14	2391485	16740395	53011252	100442377	126889889	112707070	72297141	33847581	11541621	2824822	481481	54145	3640	119	1	533233019

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-3 - 3\,x - x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + x\)

Technical details

Node id: n-f61e14215f72611a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-2ed63f4d16729a31)
Fingerprint: f6-cd2870b9fb32dc0d

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    2,     1;
  2:    5,     5,     1;
  3:   14,    21,     9,     1;
  4:   41,    82,    55,    14,     1;
  5:  122,   305,   280,   115,    20,     1;
  6:  365,  1095,  1278,   731,   210,    27,    1;
  7: 1094,  3829,  5425,  3990,  1617,   350,   35,   1;
  8: 3281, 13124, 21874, 19688, 10347,  3192,  546,  44,  1;
  9: 9842, 44289, 84888, 90426, 58518, 23571, 5796, 810, 54, 1;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = -(3*n^2-17*n+14+(4*n^2-16*n+12)*x)*P(n-1,x) + (12*n-30+(20*n-46)*x+(8*n-18)*x^2)*d/dx P(n-1,x) - (12+24*x+16*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^2 - 17*n + 14 + (4*n^2 - 16*n + 12)*x)*P[n - 1, x] + (12*n - 30 + (20*n - 46)*x + (8*n - 18)*x^2)*D[P[n - 1, x], x] - (12 + 24*x + 16*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n^2 - 17*n + 14 + (4*n^2 - 16*n + 12)*x)*v[i - 1] + (12*n - 30 + (20*n - 46)*x + (8*n - 18)*x^2)*deriv(v[i - 1], x) - (12 + 24*x + 16*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
  - ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A056241 → x ↦ x/(x+1)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003